1. The Variety of Subdivision Schemes
Kwan Pyo Ko
Department of Internet Engineering
Dongseo Univ.

2. What is Subdivision? ( S p ) = X ® S p = 2 a ¡ b j + 1
Define a smooth curves/surface as the limit of a sequence of successive refinements p j + 1 = S ( S p ) a = X b 2

3. Subdivision Surfaces

4. Why Subdivision? Arbitrary topology Multiresolution Simple code
Efficient code
Construct Wavelet

5. Chaikin’s Algorithm : converges to the quadratic B-spline. 3 p = 4 3 p+ 1 2 i = 3 4 p k + 1 2 i = 4 3
converges to the quadratic B-spline.

6. Cubic spline Algorithm :k + 1 2 i = p k + 1 2 i = 8 3 4
converges to the cubic B-spline.

7. 4-point interpolatory scheme (N. Dyn, D.Levin and J.Gregory):k + 1 2 i = p k + 1 2 i = w ( )
continuous for
C1 for j w
< 1 4
< w 1 8

8. Butterfly scheme : q = 1 2 ( p + ) w ¡ X p = [ q k e ; 3 4 8 j 5 k + 1i = [ q e

9. The mask of the butterfly scheme:

10. The mask of the butterfly scheme:

11. Ternary Subdivision Scheme:

12. where the weights are given byp j + 1 3 = i p j + 1 3 = a i b c d 2 p j + 1 3 2 = d i c b a
where the weights are given by a = 1 8 6 w ; b 3 + 2 c 7 d : C 2 f o r 1 5
< w 9

13. Subdivision Zoo Classification : stationary or non-stationary
binary or ternary
type of mesh(triangle or quadrilateral)
approximating or interpolating
linear or non-linear

14. Face split (primal type)
Vertex split (dual type)
Triangular meshes
Quad.meshes
approximating
Loop
Catmull-Clark
interpolating
Butterfly
Kobbelt ( C 2 ) ( C 2 ) ( C 1 ) ( C 1 )
Quad. meshes
approximating
Doo-Sabin , Midedge ( C 1 ) ( C 1 )

15. Binary Subdivision of B-splines
Univariate B-splines :
where : normalized B-spline of degree ( ) S r u = X i 2 Z d N ; N r r :

16. Properties : Partition of unity : Positivity : Local support :
Continuity :
Recursion : P i N r ( u ) = 1 N r ( u ) N r ( u i ) = f 2 [ ; + 1 ] N r ( u ) 2 C 1 N r ( u i ) = 1 +

17. The idea behind a Subdivision
Rewrite the curve (*) as a curve over a refined knot sequence
(*) becomes
Determine A single B-spline can be decomposed into similar B- spline of half the support. Z = 2 : S r ( u ) = X j 2 Z ^ d N ^ d r j

19. This results in where For example Chaikin’s algorithm N ( u ) = X c ¡j 2 Z c ; c r j = 2 + 1 ^ d r j = X i 2 Z c ; r = 2 ^ d 2 = X i Z c + ( 3 4 ) 1

20. Tensor Product B-spline Surfaces; s ( u ) = X i 2 Z d N
A tensor product B-spline is the product of two independently univariate B-splines, i.e N r ; s ( u i ) = v j

21. ^ d = X c c = 2 µ 1 ¶ Example 1 The mask set : r = s 2 ^ d = ¢ + ( 9 1; s j = X i 2 Z c c r ; s j = i 2 ( + ) 1
Example 1
The mask set : r = s 2 ^ d 2 ; = + ( 9 1 6 ) 3

23. Example 2
The mask for bicubic tensor product B-splines r = s 3

25. Triangular Splines ( ¤ ) S u = X d N ¡ 2 R Z
A triangular spline surface :
where
: a normalized triangular spline of degree (r+s+t-2) ( ) S r ; s t u = X i d N 2 R Z N r ; s t ( u )

27. Subdivision of Triangular Splines
The surface (*) can be rewritten over the refine grid.
A single triangular spline is decomposed into splines of identical degree over the refined grid. S r ; s t ( u ) = X j ^ d N 2 N r ; s t ( u ) = X j 2 Z c

28. ^ d = X c c = 2 X µ i ¶ N ( u ) Where
The subdivision masks for the triangular spline ^ d r ; s t j = X i 2 Z c c r ; s t j = 2 ( + ) X k i N 2 ; ( u )

29. The Doo/Sabin algorithm
(Problem)
Subdivision for tensor product quadratic B-spline surface has rigid restrictions on the topology.
Each vertex of the net must order 4.
This restriction makes the design of many surfaces difficult
Doo/Sabin presented an algorithm that eliminated this restriction by generalizing the bi-quadratic B-spline subdivision rules to include arbitrary topologies.

30. Doo-Sabin scheme P0 P1 P3 P2 P’’2 P’3 P’0 P’2 P’1 P’b P’’1 P’’3 P’’0 (\ ; 1 2 3 ) T = S D 4 S D 4 = 1 6 2 9 3 7 5
Doo-Sabin scheme

31. The subdivision masks for bi-quadratic B-spline :
Geometric view of bi-quadratic B-spline subdivision :
the new points are centroids of the sub-face formed by the face centroid, a corner vertex and the two mid-edge points next to the corner.
arbitrary topology
Generalization

32. For an n-sided face Doo/Sabin used subdivision matrix
As subdivision proceeds, the refined control point mesh becomes locally rectangular everywhere except at a fixed number of points.
Since bi-quadratic B-splines are , the surfaces generated by the Doo/Sabin algorithm are locally S D n = ( i j ) ; i j = 5 + n 4 3 2 c o s ( ) 6
extraordinay points C 1 C 1

34. The Catmull/Clark algorithm :
P42 x
P21
P31
P11
P12
P13
P14
P22
P23
P32
P43
P44
P34
P33
P24
q11
q12
q22
P41
q11 : New face points
q12 : New edge points
q22 : New vertex points

35. New face points : New edge points : New vertex points : q = P + 4 q =1 = P + 2 4 q 1 2 = C + D P ; 3 q 2 = Q 4 + R P Q = q 1 + 3 4 R = 1 4 p 2 + 3

36. The subdivision masks for bi-cubic B-spline :
Approach : generalization of bi-cubic B-spline
arbitrary topology
Generalization

37. New face points : the average of all of the old points defining the face.
New edge points : the average of the mid points of the old edge with average of the two new face.
New vertex points :
Q : the average of the new face points of all faces sharing an old vertex point.
R : the average of the midpoints of all old edges incident on the old vertex point.
S : the old vertex point. 1 4 Q + 2 R S ;

38. Modified Catmull/Clark rule
Note : tangent plane continuity was not maintained at extraordinary points.
N : order of the vertex
tangent plane continuity at extraordinary points.
Modified Catmull/Clark rule ^ S = 1 N Q + 2 R 3

40. The Loop Scheme A generalized triangular subdivision surface.
The subdivision masks for
mask A generates new control points for each vertex
mask B generates new control points for edge of the original regular triangular mesh. N 2 ; ( u ) A B

41. Q : the average of the old points that share an edge with V.
Mask B : generalization is to leave this subdivision rule intact (why?)
Mask A : The new vertex point can be computed as a convex combination of the old vertex and all old vertices that share an edge with it.
V : the old vertex point.
Q : the average of the old points that share an edge with V.
This same idea may be applied to an arbitrary triangular mesh. ^ V = 5 8 + 3 Q ;

42. Note : tangent plane continuity is lost at the extraordinary points.

43. curvature continuity at regular point.
Loop scheme :
where
curvature continuity at regular point.
tangent plane continuity at extaordinary point. ^ V = N + ( 1 ) Q ; N = 3 8 + 1 4 c o s 2

44. Mid-Edge Scheme P0 P1 P3 P2 P’3 P’0 P’2 P’1 P’’0 P’’1 P’’3 P’’2 ( P \\ ; 1 2 3 ) T = S M E 4 S M E 4 = 1 2 6 3 7 5
For an n-sided face, subdivision matrix : S M E n = ( i j ) ; i j = 1 + c o s 2 ( ) n

45. Circulant matrix
eigenvalues of can be calculated by evaluating the polynomial. S G n = 2 6 4 a 1 . 3 7 5 S G n P n ( z ) = a + 1 ; z = w j ; e i 2 n 1

46. Further Subdivision Schemes
Non-uniform scheme.
Shape preserving scheme.
Hermite-type scheme.
Variational scheme.
Quasi-linear scheme.
Poly-scale scheme.
Non-stationary scheme.
Reverse subdivision scheme.

47. Convexity Preserving ISS
A constructive approach is used to derive convexity preserving subdivision scheme.
interpolatory.
local : four points scheme.
define the first and second differences. f k + 1 2 i = F ; d f i = + 1 2

48. 3. invariant under addition of affine functions. (★)k + 1 2 i = F ( ) ; d
3. invariant under addition of affine functions.
(★) f k + 1 2 i = F ( d ; )
: subdivision function F
4. continuous
5. homogeneous, i.e. ,
6. symmetric F ( x ; y ) =

49. Note : if F=0 linear SS. only (Question)
Theorem 1 (Convexity)
A subdivision scheme of type (*) satisfying conditions 1 to 6 is convex preserving for all convex data F satisfies
Note : if F=0 linear SS. only
(Question)
under what conditions, SS(*) with conditions 1 to 6 and convexity condition generate continuously differentiable limit functions.
Answer F ( x ; y ) 1 4 m i n f g 8 C

50. Under the same conditions hold as in Theorem 1. The scheme given by
Theorem 2 (Smoothness)
Under the same conditions hold as in Theorem 1.
The scheme given by
continuously differentiable function which is convex and interpolate the data
Theorem 3 (Approximation order)
The convexity preserving subdivision scheme has approximation order 4. f k + 1 2 i = 4 d